Suppose the map \(F\)  depends smoothly on a small parameter \(\epsilon \) . We point out that all objects defined thus far may vary with \(\epsilon \) , but, to simplify notation, we do not explicitly write \(\epsilon \)  as a subscript or argument. Indeed, as \(\epsilon \)  varies, the perturbed (\(\epsilon >0\) ) normally hyperbolic invariant manifold can be written as a graph over the unperturbed (\(\epsilon =0\) ) manifold as a result of Fenichel theory, and so we can continue to use the coordinates from the original unperturbed manifold [1]}, [2]}, [3]}.
